scholarly journals Generalized cline’s formula for g-Drazin inverse in a ring

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2573-2583
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Banach Algebra ◽  
Spectral Property ◽  
Banach Spaces ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear ◽  
Group Inverses ◽  
Cline’S Formula ◽  
Generalized Drazin Inverse ◽  
Weaker Conditions

In this paper, we give a generalized Cline?s formula for the generalized Drazin inverse. Let R be a ring, and let a, b, c, d ? R satisfying (ac)2 = (db)(ac), (db)2 = (ac)(db), b(ac)a = b(db)a, c(ac)d = c(db)d. Then ac ? Rd if and only if bd ? Rd. In this case, (bd)d = b((ac)d)2d: We also present generalized Cline?s formulas for Drazin and group inverses. Some weaker conditions in a Banach algebra are also investigated. These extend the main results of Cline?s formula on g-Drazin inverse of Liao, Chen and Cui (Bull. Malays. Math. Soc., 37(2014), 37-42), Lian and Zeng (Turk. J. Math., 40(2016), 161-165) and Miller and Zguitti (Rend. Circ. Mat. Palermo, II. Ser., 67(2018), 105-114). As an application, new common spectral property of bounded linear operators over Banach spaces is obtained.

Generalized Cline’s formula and common spectral property

2020 ◽  
pp. 2150094
Author(s):  
Huanyin Chen ◽  
Marjan Sheibani Abdolyousefi
Keyword(s):  
Spectral Property ◽  
Banach Spaces ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

Let [Formula: see text] be an associative ring with an identity and suppose that [Formula: see text] satisfy [Formula: see text] If [Formula: see text] has generalized Drazin (respectively, p-Drazin, Drazin) inverse, we prove that [Formula: see text] has generalized Drazin (respectively, p-Drazin, Drazin) inverse. In particular, as applications, we obtain new common spectral property of bounded linear operators over Banach spaces.

scholarly journals A generalized Drazin inverse

1996 ◽  
Vol 38 (3) ◽  
pp. 367-381 ◽  
Author(s):  
J. J. Koliha
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Banach Algebra ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Main Theme ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Spectral Point ◽  
Generalized Drazin Inverse

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

scholarly journals Uniqueness of the maximal ideal of operators on the ℓp-sum of ℓ∞n (n ∈ ) for 1 < p < ∞

2016 ◽  
Vol 160 (3) ◽  
pp. 413-421 ◽  
Author(s):  
TOMASZ KANIA ◽  
NIELS JAKOB LAUSTSEN
Keyword(s):  
Banach Space ◽  
Recent Result ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Maximal Ideal ◽  
American Mathematical Society ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unique Maximal Ideal

AbstractA recent result of Leung (Proceedings of the American Mathematical Society, 2015) states that the Banach algebra ℬ(X) of bounded, linear operators on the Banach space X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓ1 contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces X = (⊕n∈$\mathbb{N}$ ℓ∞n)ℓp and X = (⊕n∈$\mathbb{N}$ ℓ1n)ℓp whenever p ∈ (1, ∞).

Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1

2019 ◽  
Vol 12 (05) ◽  
pp. 1950084
Author(s):  
Anuradha Gupta ◽  
Ankit Kumar
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Spectral Properties ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

Let [Formula: see text] and [Formula: see text] be two bounded linear operators on a Banach space [Formula: see text] and [Formula: see text] be a positive integer such that [Formula: see text] and [Formula: see text], then [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] have some common spectral properties. Drazin invertibility and polaroidness of these operators are also discussed. Cline’s formula for Drazin inverse in a ring with identity is also studied under the assumption that [Formula: see text] for some positive integer [Formula: see text].

On the Generalized Drazin Inverse of the Sum of Two Operators

2013 ◽  
Vol 846-847 ◽  
pp. 1286-1290
Author(s):  
Shi Qiang Wang ◽  
Li Guo ◽  
Lei Zhang
Keyword(s):  
Banach Space ◽  
Explicit Representation ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Drazin Inverse

In this paper, we investigate additive properties for the generalized Drazin inverse of bounded linear operators on Banach space . We give explicit representation of the generalized Drazin inverse in terms of under some conditions.

scholarly journals Further results on common properties of the products ac and bd

2020 ◽  
Vol 55 (2) ◽  
pp. 267-276
Author(s):  
Qingping Zeng ◽  
◽  
Kai Yan ◽  
Zhenying Wu ◽  
◽  
...  
Keyword(s):  
Banach Algebra ◽  
Banach Spaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

In this paper, we continue to investigate common properties of the products ac and bd in various categories under the assumption acd=dbd and dba=aca. These properties include generalized strongly Drazin invertibility and generalized Hirano invertibility in rings, abstract index of Fredholm elements and B-Fredholm elements in the Banach algebra context, complementability of kernels and ranges for bounded linear operators on Banach spaces.

scholarly journals The perturbation theory for the Drazin inverse and its applications II

2001 ◽  
Vol 70 (2) ◽  
pp. 189-198 ◽  
Author(s):  
Vladimir Rakočevič ◽  
Yimin Wei
Keyword(s):  
Banach Space ◽  
Perturbation Theory ◽  
Banach Algebras ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Generalized Drazin Inverse

AbstractWe study the perturbation of the generalized Drazin inverse for the elements of Banach algebras and bounded linear operators on Banach space. This work, among other things, extends the results obtained by the second author and Guorong Wang on the Drazin inverse for matrices.

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